CMS/SMC
Canadian Mathematical Society
www.cms.math.ca
Canadian Mathematical Society
  location:  PublicationsjournalsCMB
Publications        
Abstract view

The Essential Spectrum of the Essentially Isometric Operator

  Published:2012-07-16
 Printed: Mar 2014
  • H. S. Mustafayev,
    Yuzuncu Yıl University, Faculty of Science, Department of Mathematics, 65080, VAN-TURKEY
Format:   LaTeX   MathJax   PDF  

Abstract

Let $T$ be a contraction on a complex, separable, infinite dimensional Hilbert space and let $\sigma \left( T\right) $ (resp. $\sigma _{e}\left( T\right) )$ be its spectrum (resp. essential spectrum). We assume that $T$ is an essentially isometric operator, that is $I_{H}-T^{\ast }T$ is compact. We show that if $D\diagdown \sigma \left( T\right) \neq \emptyset ,$ then for every $f$ from the disc-algebra, \begin{equation*} \sigma _{e}\left( f\left( T\right) \right) =f\left( \sigma _{e}\left( T\right) \right) , \end{equation*} where $D$ is the open unit disc. In addition, if $T$ lies in the class $ C_{0\cdot }\cup C_{\cdot 0},$ then \begin{equation*} \sigma _{e}\left( f\left( T\right) \right) =f\left( \sigma \left( T\right) \cap \Gamma \right) , \end{equation*} where $\Gamma $ is the unit circle. Some related problems are also discussed.
Keywords: Hilbert space, contraction, essentially isometric operator, (essential) spectrum, functional calculus Hilbert space, contraction, essentially isometric operator, (essential) spectrum, functional calculus
MSC Classifications: 47A10, 47A53, 47A60, 47B07 show english descriptions Spectrum, resolvent
(Semi-) Fredholm operators; index theories [See also 58B15, 58J20]
Functional calculus
Operators defined by compactness properties
47A10 - Spectrum, resolvent
47A53 - (Semi-) Fredholm operators; index theories [See also 58B15, 58J20]
47A60 - Functional calculus
47B07 - Operators defined by compactness properties
 

© Canadian Mathematical Society, 2014 : https://cms.math.ca/